Time-Dependent Modelling of Pulsar Wind Nebulae

33ND INTERNATIONAL COSMIC RAY CONFERENCE,RIODE JANEIRO 2013 THE ASTROPARTICLE PHYSICS CONFERENCE

Time-dependent modelling of pulsar wind nebulae

M.J.VORSTER1,O.TIBOLLA2,S.E.S.FERREIRA1, AND S.KAUFMANN3 1 Centre for Space Research, North-West University, Potchefstroom Campus, 2520 Potchefstroom, South Africa 2 Institut fur¨ Theoretische Physik und Astrophysik, Universitat¨ Wurzburg,¨ Campus Hubland Nord, Emil-Fischer-Str. 31, D-97074 Wurzburg,¨ Germany 3 Landessternwarte, Universitat¨ Heidelberg, Konigstuhl¨ 12, D-69117 Heidelberg, Germany [email protected]

Abstract: A spatially-independent model that can be used to calculate the temporal evolution of the electron/positron spectrum in a spherically expanding pulsar wind nebula is presented. The model is applied to the young nebula G21.5-0.9, as well as to the unidentified TeV sources HESS J1427-608 and HESS J1507-622. The parameters derived from the model strengthens the idea that the unidentified sources can be identified as evolved pulsar wind nebulae.

Keywords: ISM: individual objects (G21.5-0.9, HESS J1427-608, HESS J1507-622) - ISM: supernova remnants - radiation mechanisms: non-thermal

1 Introduction 2 The Model It is well known that the X-ray synchrotron emission ob- The temporal evolution of the electron spectrum in a PWN served from pulsar wind nebulae (PWN) is produced by can be calculated using the equation (e.g., [8]) a young population of electrons, as these particles have a [ ] ∂Ne(Ee,t) ∂ relatively short lifetime. By contrast, the electrons produc- = Q(Ee,t) + E˙(Ee,t)Ne(Ee,t) , (1) ing very high energy (VHE) gamma-ray emission through ∂t ∂E the inverse Compton (IC) scattering of background pho- where Ee represents the electron energy and Ne(Ee,t) the tons have a much longer lifetime, implying that the ob- number of electrons per energy interval. The number of served TeV emission from PWNe is produced by particles electrons injected into the PWN at the termination shock, that have accumulated over the lifetime of the pulsar [1]. per time and energy interval, is given by Q(Ee,t), while the For a PWN with an average magnetic field of B = 5 µ G, second term on the right-hand side of (1) describes contin- the lifetime of an electron emitting 1keV X-rays is ∼ 3kyr, uous energy losses (or gains) suffered by the particles. The whereas the corresponding lifetime of an electron produc- value E˙(Ee,t) represents the total energy loss rate as a re- ing 1TeV gamma-rays is ∼ 19kyr [1]. sult of the various processes. Although PWNe are often identified based on the char- Emulating [9], a broken power-law spectrum is used to acteristics of their radio and X-ray synchrotron spectra model the emission from the sources studied in this paper (e.g., [1, 2]), it has been proposed by [3] that the magnetic { field in an evolved PWN could evolve below the value of QR (Eb/Ee), if Emin ≤ Ee ≤ Eb Q(Ee,t) = 2 , (2) the interstellar medium (ISM), resulting in these sources QX (Eb/Ee) , if Eb < Ee ≤ Emax being undetectable at synchrotron frequencies. However, due to the longer lifetimes of the VHE gamma-ray produc- where QR and QX are normalisation constants, Emin and ing electrons, these ancient PWNe may still be visible at Emax the minimum and maximum electron energy respec- TeV energies. As PWNe form a significant fraction of de- tively, and Eb the energy where the spectrum transitions be- tected TeV sources, the ancient PWN scenario could offer tween the two components. With the results of [10] taken an explanation for a number of unidentified TeV sources into account, it is not an a priori requirement that the two [4] that lack a clear synchrotron counterpart. components should have the same intensity at Eb. In this paper we present time-dependent modelling of The normalisation constants are determined by the pre- scription that the total energy in a given source component the candidate PWNe HESS J1427-608 [4] and HESS η J1507-622 [5]. The aim is to not only investigate the an- should be some fraction i (i = R,X) of the pulsar’s spin- ( ) cient PWN hypothesis, but also to determine whether a down luminosity L t [9] ∫ clear argument can be made for identifying HESS J1427- pi η 608 and HESS J1507-622 as PWNe. Before presenting the Qi (Eb/Ee) EedEe = iL(t), (3) modelling results, the spatially-independent model used to with calculate the temporal evolution of the electron spectrum L is introduced. In order to test the model, it is first applied 0 L(t) = 2 . (4) to the young PWN G21.5-0.9. (1 +t/τ) For a more extended discussion of the results, [6] should In the expression above L0 is the initial luminosity and τ be consulted. Using the present model, predictions for ad- the characteristic spin-down time scale of the pulsar, while ditional unidentified TeV sources have also been made by it is assumed that the pulsar is a pure dipole radiator with [7]. a braking index of 3. Time-dependent modelling of PWNe 33ND INTERNATIONAL COSMIC RAY CONFERENCE,RIODE JANEIRO 2013

The expressions given in [11] are used to calculate the where ηB is the fraction of the pulsar’s spin-down luminos- non-thermal energy losses that result from synchrotron ra- ity converted into magnetic energy, and Vpwn(t) the volume diation and IC scattering in the Thomson regime. During of the PWN. the modelling it was found that the energy density of the The ratio of electromagnetic to particle energy in the IC target photons becomes larger than the energy density nebula σ is defined in the model as of the magnetic field, and it thus becomes necessary to in- η clude a Klein-Nishina correction to the non-thermal energy σ = B , (8) loss rate. The approximations derived by [12] is used for ηR + ηX this purpose. Following [8], adiabatic cooling (heating) is described in the present model by with the constraint σ ≲ 1 imposed [1]. An additional constraint is supplied by ηB + ηR + ηX ≲ 1. This sum is not v (t) set strictly equal to unity to allow for the fact that a frac- E˙ (E ,t) = pwn E , (5) ad e e tion ηrad of the spin-down luminosity is radiated away in Rpwn(t) the form of pulsed emission. For the modeling, ηrad ≲ 1% is used. Note that this small value used for ηrad effectively where vpwn(t) and Rpwn(t) are respectively the expansion σ ≃ η velocity and radius of the PWN. An important parameter implies that B. Particles can also escape from the PWN as a result of in the model is the evolution of Rpwn(t). Not only is this τ required for the adiabatic energy loss rate (5), but also for diffusion. The escape time scale esc is given by the evolution of the average magnetic field and escape loss R2 (t) rate (as will be discussed below). τ = pwn , (9) Simulations predict that the evolution of a PWN can be esc 6κ(t) divided into three phases: in the initial evolutionary phase r the nebula expands at a rate Rpwn(t) ∝ t 1 (e.g., [13, 14]). where κ(t) is the diffusion coefficient. Diffusion in a PWN When the pressure in the supernova shell remnant (SNR) results from particles interacting with irregularities in the becomes small enough, the reverse shock of the shell will magnetic field, and the scaling κ(t) ∝ 1/Bpwn(t) is used in start to propagate towards the centre of the remnant [15], the model. The diffusion coefficient is also proportional to reaching the outer boundary of the PWN after a time trs, energy, i.e., κ ≡ κ(Ee/1 TeV). and the next evolutionary phase begins. The reverse shock The energy loss equation (1) is not solved directly, but compresses the PWN over a time scale of a few thou- rather using the approximation method described in [6]. sand years (e.g., [16, 17]), leading to a stronger magnetic The non-thermal emission is calculated using the appro- field and enhanced synchrotron losses [13]. After the ini- priate expressions given in [11]. This implies that Klein- tial compression, the PWN will enter a second expansion Nishina effects are taken into account when calculating the phase starting at t = tse. In contrast to the first expansion IC spectrum. phase, Rpwn(t) does not evolve smoothly, but has an oscil- latory nature [16, 17]. As a first approximation, all three evolutionary phases of Rpwn(t) are modelled using a power- 3 Results law  r  R0(t/t0) 1 if t < trs Parameter G21.5 J1427 J1507 r r Rpwn(t) = R0(trs/t0) 1 (t/trs) 2 if trs ≤ t < tse 38 −1  L0 (10 ergs ) 0.54∗ 5.5 1.2 r1 r2 r3 ≥ R0(trs/t0) (tse/trs) (t/tse) if t tse τ (kyr) 3∗ 3 0.5 (6) ∗ tage (kyr) 0.87 10 24 where R0 = 0.01 pc is the initial radius at t0 = 10 yr [14]. Bage (µ G) 230 0.4 1.7 The values r1, r2, and r3 are not linearly independent as η the size of the PWN predicted by the model must be equal R 0.68 0.81 0.8 η to the observed size. Note that the distance to the source X 0.13 0.18 0.17 η η ∗ ∗ ∗ d r r r R/ X 5.4 4.5 4.7 influences the values of 1, 2 and 3, as a larger value − of d implies a larger source, and hence a faster expansion. σ (10 3) 180 0.01 30 −3 The expansion velocity of the PWN follows from the usual Emin (10 TeV) 0.3 100 1 2 definition vpwn(t) = dRpwn(t)/dt. Emin (10 TeV) 2.7 3 2 An expression for trs has been derived by a number of Eb (TeV) 0.1 0.18 0.5 25 2 −1 authors, including [13, 18]. It is estimated that an SNR κage (10 ETeV cm s ) 2.2 7 15 with an ejecta mass of M = 5M⊙, and a kinetic energy ∗ ∗ ∗ ej κ/κBohm 390 2.3 20 51 ∗ ∗ ∗ of Eej = 10 erg, expanding into an ISM with a density d (kpc) 4.8 11 6 −3 of nism = 1 cm , will have a reverse shock time scale of t ≈ 6000 yr. Using a smaller value for the ISM density rs Table 1: Values derived with the model for the various free (n = 0.1 cm−3) increases the reverse shock time scale to ism parameters. Values marked with an * represent parameters trs ≈ 11000 yr. The evolution of the average magnetic field in the neb- that were kept fixed, or parameters that follow from the derived model parameters. The ratio κ/κ expresses the ula Bpwn(t) is calculated using the conservation of mag- Bohm netic flux [8] derived diffusion coefficient in terms of the Bohm diffusion coefficient. ∫ t 2 Bpwn(t) ηBL(t)dt = Vpwn(t) , (7) 0 8π Time-dependent modelling of PWNe 33ND INTERNATIONAL COSMIC RAY CONFERENCE,RIODE JANEIRO 2013

Figure 1: Model prediction for G21.5-0.9. Radio data Figure 2: Model prediction for the unidentified TeV source are taken from [24, 25, 26, 27, 28, 29, 30, 21, 31], HESS J1427-608. The radio data is taken from [38], the X- infra-red data from [32], X-ray data from ray data from [37], the Fermi data from [39], and the TeV [33, 34] and the INTEGRAL Science Data Centre data from [4]. The parameters listed in Table 1 are for the (http://www.isdc.unige.ch/heavens webapp/integral), and Bage = 0.4 µ G scenario. TeV data from [35]. The data were also modelled using the Maxwellian spectrum calculated by [36]. a larger magnetic field of Bage = 4 µ G. However, this scenario predicts a bright radio synchrotron nebula that has 3.1 G21.5-0.9 thus far not been observed. The fact that the model cannot simultaneously predict both the radio and X-ray obser- L = . × 37 −1 With a spin-down luminosity of 3 3 10 ergs [19], vations indicates that one of the two synchrotron sources the pulsar in the supernova remnant G21.5-0.9 is one of is not a plausible counterpart to HESS J1427-608. Note the most energetic pulsars in the Galaxy. The PWN is that the parameters listed in Table 1 are specifically for the located at a distance of 4.8kpc [20], with an estimated age B = 0.4 µ G scenario of 870yr [21]. For the modeling of G21.5-0.9, the value age τ = 3kyr is used [22]. The PWN is too young to have interacted with the reverse shock, and must therefore still 3.3 HESS J1507-622 be in the first expansion phase. Also discovered in a H.E.S.S. Galactic plane survey is the Figure 1 shows the model prediction for the non-thermal bright VHE source HESS J1507-622 [5]. This source is radiation spectra, with the derived parameters listed in Ta- unique in the sense that it lies ∼ 3.5◦ from the Galac- ble 1. From the model a present-day magnetic field of µ tic plane, whereas all other unidentified TeV sources lie Bage = 230 G is derived. This is comparable to the value within 1◦ from the Galactic equator. Most Galactic VHE µ ∼ of Bage = 300 G derived for the 1kyr old Crab nebula. sources are connected to young stellar populations (located To obtain the model prediction, the diffusion coefficient in the disk), and one would therefore not expect a bright κ × 25 2 −1 should not be larger than = 2.2 10 ETeV cm s , or VHE source at the observed position. Furthermore, the ab- κ κ in terms of the Bohm diffusion coefficient, = 390 Bohm. sence of a bright X-ray counterpart is surprising as the com- η η ◦ Furthermore, a ratio of R/ X = 5.4 is derived for the con- parably low hydrogen column density at ∼ 3.5 leads to version efficiencies. From the model prediction a value of a considerably lower absorption of X-rays, as well as re- σ = 0.18 is derived for the ratio of magnetic to particle en- duced background emission [5]. ergy, larger than the value of σ ∼ 0.003 calculated for the Assuming that HESS J1507-622 is still in the initial ex- Crab Nebula [23]. pansion phase, a break energy of Eb = 0.5TeV is derived. This is an order of magnitude larger than the value derived 3.2 HESS J1427-608 for other PWNe [42, 41]. Although it is not excluded that such a large break energy is the result of shock accelera- One of the unidentified TeV sources discovered in a tion, an alternative scenario is favoured in the present pa- H.E.S.S. galactic plane survey is HESS J1427-608 [4]. per where HESS J1507-622 has been compressed by the From the model prediction an initial luminosity of reverse shock of the SNR. As E˙ /E is constant in (5), the × 38 −1 e e L0 = 5.5 10 ergs is derived, leading to a present- effect of adiabatic losses is to shift the electron spectrum × 37 −1 day luminosity of L = 2.9 10 ergs . The ratios to lower energies without affecting the spectral shape. Dur- η η σ −5 R/ X = 4.5 and = 10 are derived, along with a ing the compression phase, the exact opposite will occur. µ present-day magnetic field of Bage = 0.4 G. Lastly, a dif- The particles will be subjected to adiabatic heating, caus- 25 2 −1 fusion coefficient of κ = 5.6 × 10 ETeV cm s is pre- ing the electron spectrum to shift to higher energies, lead- κ κ dicted by the model, or equivalently, = 2.3 Bohm. While ing to an increase in the value of Eb. these parameters may lead to an acceptable agreement be- For the compression scenario, the value of trs = 20kyr tween the model and radio data, Figure 2 shows that this is used, while the compression phase is chosen to last for scenario significantly under-predicts the Suzaku spectrum 4kyr. It is assumed that the nebula has not yet entered [37]. It is possible to predict the X-ray observations using the second expansion phase, so that the current age of the Time-dependent modelling of PWNe 33ND INTERNATIONAL COSMIC RAY CONFERENCE,RIODE JANEIRO 2013

between the two components cannot be an artefact of PWN evolution. A characteristic of the discontinuous spectrum is that a particle conversion efficiency must be specified for both the low (ηR) and high-energy (ηX) components, with a ratio of ηR/ηX ∼ 4.5 − 5.4 derived for the three sources. The research presented in this paper is discussed in more detail by [6], with additional modelling results of unidentified TeV sources presented by [7].

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